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ISSN 1472-2739 (on-line) 1472-2747 (printed)
Algebraic & Geometric Topology
Volume 3 (2003) 587–591
Published: 21 June 2003
ATG
What is a virtual link?
Greg Kuperberg
Abstract Several authors have recently studied virtual knots and links
because they admit invariants arising from R-matrices. We prove that
every virtual link is uniquely represented by a link L ⊂ S ×I in a thickened,
compact, oriented surface S such that the link complement (S × I) \ L has
no essential vertical cylinder.
AMS Classification 57M25; 57M27 57M15
Keywords Virtual link, tetravalent graph, stable equivalence
A virtual link L is an equivalence class of decorated, finite, tetravalent graphs Γ.
The edges at each vertex must be cyclically ordered, and two opposite edges are
labelled as an overcrossing, while the other two are labelled as an undercrossing.
The equivalence relation is the one given by Reidemeister moves. The notion
was proposed by Kauffman [5] in light of the fact that R-matrices and quandles,
which are commonly used to make link invariants, also yield invariants of virtual
links.
We will borrow from the analysis of virtual links by Carter, Kamada, and
Saito [1]. They show that virtual links are equivalent to stable equivalence
classes of links projections onto compact, oriented surfaces. (Fenn, Rourke, and
Sanderson have written a self-contained proof [2].) The surface S need not be
connected, but we require that each component contains at least one component
of the projection P . The projection P is considered up to Reidemeister moves,
and the stabilization operation consists of adding a handle to S . Note that the
feet of a stabilizing handle may lie in any two regions of the complement of P
(Figure 1).
As Figure 1 also shows, the reverse destabilization operation consists of cutting
the surface S along a circle C which is disjoint from P , and capping the resulting boundary. If C separates the connected component of S 0 of S containing
it, then we require that both components of S 0 \ C contain part of P ; otherwise
destabilization would create a naked surface component.
It is also well-known that a link drawn on a surface S , considered up to Reidemeister moves, is equivalent to a (tame) link L ⊂ S × I in the thickened surface
c Geometry & Topology Publications
588
Greg Kuperberg
Figure 1: The stabilization move on S × I .
S , considered up to isotopy. The destabilization operation, then, consists of
cutting S along a vertical annulus C × I which is disjoint from L, and capping
the two resulting annuli with thickened disks D 2 ×I . Since L is only considered
up to isotopy, it is equivalent to allow this operation with any topologically vertical annulus A, i.e., any properly embedded annulus isotopic to some C × I .
This will be our working definition of virtual links.
Theorem 1 Every stable equivalence class of links in thickened surfaces has
a unique irreducible representative.
Theorem 1 is proved in the same way as several classical results in 3-manifold
topology, from the unique factorization of knots and 3-manifolds [8, 7] to the
Jaco-Shalen-Johannson theorem [3, 4], from which Theorem 1 also follows as a
corollary. Another similar result is the author’s classification of scalar involutory
Hopf words [6]. The motivation is also similar, since a virtual link can be viewed
as a scalar R-matrix word.
Proof In outline, we induct on the complexity of intersection between two
destabilization annuli by compressing one along an innermost disk of the other.
First generalize the definition of destabilization of a link L ⊂ S × I to include
two other operations:
Algebraic & Geometric Topology, Volume 3 (2003)
What is a virtual link?
589
(1) If a sublink L0 ⊂ L is separated from the rest of L by a sphere or disk
A ⊂ S × I , then we can remove L0 from S × I and place it in a separate
thickened sphere S 2 × I .
(2) If an annulus A divides S × I with L entirely on one side and some genus
of S on the other side, we can cut S × I along A, discard the naked component,
and cap the remaining component. Both operations can easily be reproduced
by destabilization as defined previously.
Say that a surface is admissible if it is a vertical annulus, a sphere, or a proper
disk; and that an admissible surface is essential if it does not bound a ball in
(S × I) \ L. Thus, admissible, essential surfaces are those along which we can
destabilize L ⊂ S × I .
Suppose, to the contrary of the conclusion, that some link L ⊂ S × I has more
than one irreducible descendant. If S has c components with total genus g , and
L has n components, assume that g + n − c is minimal among counterexamples.
(Note that n ≥ c.) Then every destabilization of L ⊂ S × I has a unique
irreducible descendant, since destabilization always reduces g + n − c. Say that
two such destabilizations, L ⊂ S1 × I and L ⊂ S2 × I , are descent equivalent
if their irreducible descendants are isomorphic. The aim is to show that all
destabilizations of L ⊂ S × I are descent equivalent.
For example, if A1 and A2 are disjoint admissible, essential surfaces, then the
resulting destabilizations L ⊂ S1 × I and L ⊂ S2 × I are descent equivalent.
This is immediate if A1 and A2 are parallel. If they are not, then we can
destabilize each L ⊂ Si × I along A3−i to produce a common descendant.
Suppose that A1 and A2 are descent-inequivalent surfaces in general position,
and that they intersect in the fewest curves among descent-inequivalent pairs
in (S × I) \ L. If a curve C ⊂ A1 ∩ A2 is a circle, then it is either horizontal in
Ai , if Ai is an annulus and C is parallel to ∂Ai , or it bounds a disk in Ai . If
C is an arc, then it is either vertical, if Ai is an annulus and C connects the
two components of ∂Ai , or it and part of the boundary of Ai bound a disk.
If a circle of A1 ∩ A2 is non-horizontal in A1 (say), then some such circle C is
innermost, meaning that it bounds a naked disk D in A1 , as in Figure 2(a).
In this case let A02 and A002 be the connected components of the compression of
A2 along the disk D, as in Figure 2(b). Both A02 and A002 are admissible, and
at least one is essential, for otherwise A2 would not be. If A02 (say) is essential,
then it intersects A1 less than A2 does, and it does not intersect A2 at all. But
since A1 and A2 intersect least among descent-inequivalent pairs of essential
surfaces, it would follow that A1 and A2 are both descent-equivalent to A02 , a
contradiction.
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Greg Kuperberg
A2
A2
A1
C
A1
A002
D
(a)
A02
(b)
Figure 2: Compressing A2 along the disk D to simplify A1 ∩ A2 (side view).
The same argument applies if A1 ∩ A2 has a non-vertical arc in A1 , and of
course also to non-horizontal circles and non-vertical arcs in A2 . Thus A1 ∩ A2
consists entirely of vertical segments or horizontal circles in both A1 and A2 .
In particular, both A1 and A2 are vertical annuli and not disks or spheres.
Suppose that A1 ∩ A2 consists of horizontal circles. We can assume that none
of the four circles of ∂A1 and ∂A2 bounds a disk in S × ∂I . If, say, C ⊂ A1
bounds a naked disk D ⊂ S × ∂I , then we replace A1 by the disk D ∪ A1 and
reduce to a previous case without worsening A1 ∩A2 . Otherwise let C ⊂ A1 ∩A2
be an outermost circle, meaning that it and one component of ∂A1 bound a
naked annulus A ⊂ A1 . The circle C divides A2 into two annuli A02 and A002 ,
one of which, say A02 , makes a vertical annulus together with A. The annulus
A ∪ A02 is necessarily essential since its boundary circles do not bound disks in
S × ∂I . But after displacement, A ∪ A02 intersects A1 and A2 less than they
do each other. It is therefore descent equivalent to both.
Finally suppose that A1 ∩A2 consists of vertical arcs. The boundary of a regular
neighborhood of A1 ∪ A2 consists of vertical annuli B1 , B2 , . . . , Bn . Each Bi
is disjoint from both A1 and A2 , so if any of them is essential, it is descent
equivalent to both A1 and A2 , a contradiction. But if they are all inessential,
then one of them, say B1 , separates A1 and A2 from the link L and bounds
a ball that contains A1 and A2 . This contradicts the hypothesis that A1 and
A2 are essential.
Algebraic & Geometric Topology, Volume 3 (2003)
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What is a virtual link?
Theorem 1 implies that if two links are equivalent as virtual links, then they
are equivalent as links. It also generalizes to oriented virtual links, to colored
virtual links, and even to virtual tangled graphs. The proof in each case is the
same.
Acknowledgments
The author would like to thank Dror Bar-Natan, Louis Kauffman, Colin Rourke,
and Dylan Thurston for bringing the virtual link problem to his attention, and
for very helpful discussions.
The author was supported by NSF grant DMS #0072342.
References
[1] J. Scott Carter, Seiichi Kamada, and Masahico Saito, Stable equivalence of
knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications
11 (2002), no. 3, 311–322, arXiv:math.GT/0008118.
[2] Roger Fenn, Colin Rourke, and Brian Sanderson, The rack space,
arXiv:math.GT/0304228.
[3] William H. Jaco and Peter B. Shalen, Seifert-fibered spaces in 3-manifolds, Mem.
Amer. Math. Soc. 21 (1979), no. 220, viii+192.
[4] Klaus Johannson, Homotopy equivalences of 3-manifolds with boundaries,
Springer, Berlin, 1979.
[5] Louis H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7,
663–690, arXiv:math.GT/9811028.
[6] Greg Kuperberg, Involutory Hopf algebras and 3-manifold invariants, Internat.
J. Math. 2 (1991), no. 1, 41–66, arXiv:math.QA/9201301.
[7] J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. Math. 84
(1962), 1–7.
[8] Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57–104.
Department of Mathematics, University of California
Davis, CA 95616, USA
Email: greg@math.ucdavis.edu
Received: 18 August 2002
Revised: 15 June 2003
Algebraic & Geometric Topology, Volume 3 (2003)